Object is hollow with boolean modifier
I know I'm 6 months late for this but I would just like to share an answer since I was also currently doing Andrew's tutorial for the Anvil and I also encountered this problem. I think the issue here is that after deleting the half of the model, you basically declared to Blender that your particular model is open and empty inside and that is why whenever you apply the Mirror modifier, you end up with a hollow model since the base model was hollow in the first place.
The solution I found for this is after deleting the half of the anvil, select all the vertices that were left and press F to create a new face.


Then do the rest with the Boolean modifier and the model should be solid inside. And also make sure that the Boolean modifier is on top of the Mirror modifier to see the cut on both sides. Hope this helps :)